**Fig. 2** Coordinate systems and corresponding coordinate rotations based on traditional coordinate system for Gaussian beam reflection (TCS) and novel coordinate system for Gaussian beam reflection (NCS) in four equal-sided non-planar ring resonators (NPRO), β: folding angle, m_{1} and m_{2}: spherical mirrors with radius of R_{1} and R_{2}, m_{3} and m_{4}: planar mirrors, A_{j}(j = 1,2,3,4): incident angles on four mirrors, P_{j}(j = 1,2,3,4): terminal points of the resonator, P_{e}, P_{f}, P_{g}, P_{h}, O_{1}, O_{2}: the midpoints of straight lines P_{1}P_{2}, P_{2}P_{3}, P_{3}P_{4}, P_{4}P_{1}, P_{1}P_{3} and P_{2}P_{4} separately, φ_{tj}(j = 1,2,3,4) and φ_{j}(j = 1,2,3,4): coordinate rotation angles based on TCS and NCS respectively, n_{j}(j = 1,2,3,4): the binormals at points P_{j}(j = 1,2,3,4), (x_{tj}, y_{j}, z_{j}) and (x_{j}, y_{j}, z_{j})(j = 1,2,3,4): coordinate systems for the incident beam (based on TCS and NCS respectively) before being reflected from points P_{j}(j = 1,2,3,4), (x_{tjr}, y_{jr}, z_{jr}) and (x_{jr}, y_{jr}, z_{jr})(j = 1,2,3,4): coordinate systems for the reflected beam (based on TCS and NCS respectively) after being reflected from points P_{j}(j = 1,2,3,4), δ_{jz}(j = 1,2,3,4): axial displacement of mirrors m_{j}(j = 1, 2, 3, 4), δ_{jx}, δ_{jy}(j = 1,2): radial displacements of the spherical mirrors m_{1} and m_{2}. (Note: The positive directions of y_{j} and y_{jr}(j = 1,2,3,4) are along the directions of n_{j}(j = 1,2,3,4); the positive directions of z_{j} and z_{jr}(j = 1,2,3,4,b,c) are along the direction of beam propagation; (x_{t1}, x_{t1r}, x_{1}, x_{1r}), (x_{t2}, x_{t2r}, x_{2}, x_{2r}), (x_{t3}, x_{t3r}, x_{3}, x_{3r}) and (x_{t4}, x_{t4r}, x_{4}, x_{4r}) are located at the incident planes of P_{4}P_{1}P_{2}, P_{1}P_{2}P_{3}, P_{2}P_{3}P_{4} and P_{3}P_{4}P_{1} separately; the positive directions of δ_{1x}, δ_{2x}, δ_{1z}, δ_{2z}, δ_{3z} and δ_{4z} are along the directions of straight lines P_{2}P_{4}, P_{1}P_{3}, P_{1}O_{2}, P_{2}O_{1}, P_{3}O_{2} and P_{4}O_{1} separately; the positive direction of δ_{jy} (j = 1,2) is along the direction of n_{j}(j = 1,2).)